Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $a \neq 0$. $k = \dfrac{6a - 48}{a - 6} \times \dfrac{a^2 - 3a - 18}{3a^2 - 24a} $
Explanation: First factor the quadratic. $k = \dfrac{6a - 48}{a - 6} \times \dfrac{(a - 6)(a + 3)}{3a^2 - 24a} $ Then factor out any other terms. $k = \dfrac{6(a - 8)}{a - 6} \times \dfrac{(a - 6)(a + 3)}{3a(a - 8)} $ Then multiply the two numerators and multiply the two denominators. $k = \dfrac{ 6(a - 8) \times (a - 6)(a + 3) } { (a - 6) \times 3a(a - 8) } $ $k = \dfrac{ 6(a - 8)(a - 6)(a + 3)}{ 3a(a - 6)(a - 8)} $ Notice that $(a - 8)$ and $(a - 6)$ appear in both the numerator and denominator so we can cancel them. $k = \dfrac{ 6\cancel{(a - 8)}(a - 6)(a + 3)}{ 3a\cancel{(a - 6)}(a - 8)} $ We are dividing by $a - 6$ , so $a - 6 \neq 0$ Therefore, $a \neq 6$ $k = \dfrac{ 6\cancel{(a - 8)}\cancel{(a - 6)}(a + 3)}{ 3a\cancel{(a - 6)}\cancel{(a - 8)}} $ We are dividing by $a - 8$ , so $a - 8 \neq 0$ Therefore, $a \neq 8$ $k = \dfrac{6(a + 3)}{3a} $ $k = \dfrac{2(a + 3)}{a} ; \space a \neq 6 ; \space a \neq 8 $